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In this paper, we establish a q-integral formula by using the orthogonality relation, and also provide a new proof of the q-orthogonality relation for the continuous q-ultraspherical polynomials. A new q-beta integral with five parameters is evaluated. [ABSTRACT FROM AUTHOR]

In this paper, we consider the weak limit of the normalized measure for Askey–Wilson polynomials when the parameter q approaches - 1 from the right. We use two different methods to prove that the weak limit is a discrete measure with two mass points that are symmetric about the origin. The weights on these two mass points are, however, not always the same. We also calculate the weak limit of the q-ultraspherical measure when q approaches a complex root of unity. [ABSTRACT FROM AUTHOR]

Starting from the moment sequences of classical orthogonal polynomials we derive the orthogonality purely algebraically. We consider also the moments of ( q = 1 ) classical orthogonal polynomials, and study those cases in which the exponential generating function has a nice form. In the opposite direction, we show that the generalized Dumont–Foata polynomials with six parameters are the moments of rescaled continuous dual Hahn polynomials. Finally, we show that one of our methods can be applied to deal with the moments of Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

We study the specializations of parameters in Koornwinder polynomials to obtain Macdonald polynomials associated to the subsystems of the affine root system of type (C n ∨ , C n) in the sense of Macdonald (Affine Hecke algebras and orthogonal polynomials, Cambridge tracts in mathematics, Cambridge Univ Press, 2003), and summarize them in what we call the specialization table. As a verification of our argument, we check the specializations to type B, C and D via Ram–Yip type formulas of non-symmetric Koornwinder and Macdonald polynomials. [ABSTRACT FROM AUTHOR]

In this paper we show that the only sequences of orthogonal polynomials (P n) n ≥ 0 satisfying ϕ (x) D q P n (x) = a n S q P n + 1 (x) + b n S q P n (x) + c n S q P n - 1 (x) , ( c n ≠ 0 ) where ϕ is a well chosen polynomial of degree at most two, D q is the Askey-Wilson operator and S q the averaging operator, are the multiple of Askey-Wilson polynomials, or specific or limiting cases of them. [ABSTRACT FROM AUTHOR]

The purpose of this note is to characterize all the sequences of orthogonal polynomials (P n) n ≥ 0 such that ▵ ▵ x (s - 1 / 2) P n + 1 (x (s - 1 / 2)) = c n (▵ + 2 I) P n (x (s - 1 / 2)) , where I is the identity operator, x defines a class of lattices with, generally, nonuniform step-size, and ▵ f (s) = f (s + 1) - f (s) . The proposed method can be applied to similar and to more general problems involving the mentioned operators, in order to obtain new characterization theorems for some specific families of classical orthogonal polynomials on lattices. [ABSTRACT FROM AUTHOR]

In this paper, we review the properties of six families of orthogonal polynomials that form the main bulk of the collection called the Askey–Wilson scheme of polynomials. We give connection coefficients between them as well as the so-called linearization formulae and other useful important finite and infinite expansions and identities. An important part of the paper is the presentation of probabilistic models where most of these families of polynomials appear. These results were scattered within the literature in recent 15 years. We put them together to enable an overall outlook on these families and understand their crucial rôle in the attempts to generalize Gaussian distributions and find their bounded support generalizations. This paper is based on 65 items in the predominantly recent literature. [ABSTRACT FROM AUTHOR]

A proof of the orthogonality relation for the Askey–Wilson polynomials is given by using a generating function for the Askey–Wilson polynomials and the uniqueness of a rational function expansion. We further use the orthogonality relation for the Askey–Wilson polynomials and a q-series transformation formula to evaluate a general q-beta integral with eight parameters. The integrand of this q-beta integral is the product of two terminating 5φ4 series and the value is a 10φ9 series. [ABSTRACT FROM AUTHOR]

We use the Poisson kernel of the continuous q -Hermite polynomials to introduce families of integral operators. One of them is semigroups of linear operators. We describe the eigenvalues and eigenfunctions of one family of operators. The action of the semigroups of operators on the Askey–Wilson polynomials is shown to only change the parameters but preserves the degrees, hence we produce transmutation relations for the Askey–Wilson polynomials. The transmutation relations are then used to derive bilinear generating functions involving the Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

We examine the properties of distributions with the density of the form: 2 A n c n - 2 c 2 - x 2 π ∏ j = 1 n (c (1 + a j 2 ) - 2 a j x) , where c, a1, ..., an are some parameters and An a suitable constant. We find general forms of An, of k-th moment and of k-th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so called Askey–Wilson scheme. On the way we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters a1, ..., an forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases n = 2, 4, 6. [ABSTRACT FROM AUTHOR]

Recently, it was observed that the roots of the Askey-Wilson polynomial are retrieved at the unique global minimum of an associated strictly convex Morse function [J. F. van Diejen and E. Emsiz, Lett. Math. Phys. 109 (2019), pp. 89-112]. The purpose of the present note is to infer that the corresponding gradient flow converges to the roots in question at an exponential rate. [ABSTRACT FROM AUTHOR]

We study a substitute for the matrix product ansatz for asymmetric simple exclusion process with open boundary in the "singular case" α β = q N γ δ , when the standard form of the matrix product ansatz of Derrida et al. (J Phys A 26(7):1493–1517, 1993) does not apply. In our approach, the matrix product ansatz is replaced with a pair of linear functionals on an abstract algebra. One of the functionals, φ 1 , is defined on the entire algebra, and determines stationary probabilities for large systems on L ≥ N + 1 sites. The other functional, φ 0 , is defined only on a finite-dimensional linear subspace of the algebra, and determines stationary probabilities for small systems on L < N + 1 sites. Functional φ 0 vanishes on non-constant Askey–Wilson polynomials and in non-singular case becomes an orthogonality functional for the Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

The Askey-Wilson polynomials are the most general classical orthogonal polynomials that are known, and the Nassrallah-Rahman integral is a very general extension of Euler's integral representation of the classical 2F1 function. Based on a q-series transformation formula and the Nassrallah-Rahman integral we prove a q-beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This q-beta integral also allows us to derive a curious double q-series transformation formula, which includes one formula of Al-Salam and Ismail as a special case. [ABSTRACT FROM AUTHOR]

We develop analogs of exponential and trigonometric functions (including the basic exponential function) and derive their fundamental properties: addition formula, positivity, reciprocal and fundamental relations of trigonometry. We also establish a binomial theorem, characterize symmetric orthogonal polynomials and provide a formula for computing the nth-derivatives for analytic functions on nonuniform lattices (q-quadratic and quadratic variables). [ABSTRACT FROM AUTHOR]

Given an entire function f(z) that has only negative zeros, we shall prove that all the functions of type f(m)(x)/f(n)(x), m > n are completely monotonic. Examples of this type are given for Laguerre polynomials, ultraspherical polynomials, Jacobi polynomials, Stieltjes-Wigert polynomials, q-Laguerre polynomials, Askey-Wilson polynomials, Ramanujan function, qexponential functions, q-Bessel functions, Euler's gamma function, Airy function, modified Bessel functions of the first and the second kind, and the confluent basic hypergeometric series. [ABSTRACT FROM AUTHOR]

In previous work [12–14] , the first and third authors introduced staircase tableaux, which they used to give combinatorial formulas for the stationary distribution of the asymmetric simple exclusion process (ASEP) and for the moments of the Askey–Wilson weight function. The fact that the ASEP and Askey–Wilson moments are related at all is unexpected, and is due to [45] . The ASEP is a model of particles hopping on a one-dimensional lattice of N sites with open boundaries; particles can enter and exit at both left and right borders. It was introduced around 1970 [34,43] and is cited as a model for both traffic flow and translation in protein synthesis. Meanwhile, the Askey–Wilson polynomials are a family of orthogonal polynomials in one variable which sit at the top of the hierarchy of classical orthogonal polynomials. So from this previous work, we have the relationship ASEP −− staircase tableaux −− Askey–Wilson moments. The Askey–Wilson polynomials can be viewed as the one-variable case of the multivariate Koornwinder polynomials, which are also known as the Macdonald polynomials attached to the non-reduced affine root system ( C n ∨ , C n ). It is natural then to ask whether one can generalize the relationships among the ASEP, Askey–Wilson moments, and staircase tableaux, in such a way that Koornwinder moments replace Askey–Wilson moments. In [15] , we made a precise link between Koornwinder moments and the two-species ASEP , a generalization of the ASEP which has two species of particles with different “weights.” In this article we introduce rhombic staircase tableaux , and show that we have the relationship 2-species ASEP −− rhombic staircase tableaux −− Koornwinder moments. In particular, we give formulas for the stationary distribution of the two-species ASEP and for Koornwinder moments, in terms of rhombic staircase tableaux. [ABSTRACT FROM AUTHOR]

In many cases one may encounter an integral which is of q -Mellin–Barnes type. These integrals are easily evaluated using theorems which have a long history dating back to Slater, Askey, Gasper, Rahman and others. We derive some interesting q -Mellin–Barnes integrals and using them we derive transformation and summation formulas for nonterminating basic hypergeometric functions. The cases which we treat include ratios of theta functions, the Askey–Wilson moments, nonterminating well-poised ϕ 2 3 , nonterminating very-well-poised W 4 5 , W 7 8 , products of two nonterminating ϕ 1 2 's, square of a nonterminating well-poised ϕ 1 2 , a nonterminating W 9 10 , two nonterminating W 11 12 's and several nonterminating summations which arise from the Askey–Roy and Gasper integrals. [ABSTRACT FROM AUTHOR]

We prove a general expansion formula in Askey–Wilson polynomials using Bailey transform and Bressoud inversion. As applications, we give new proofs and generalizations of some recent results of Ismail–Stanton and Liu. Moreover, we prove a new q -beta integral formula involving Askey–Wilson polynomials, which includes the Nassrallah–Rahman integral as a special case. We also give a bootstrapping proof of Ismail–Stanton's recent generating function of Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.

Formulae expressing explicitly the q-difference derivatives and the moments of the polynomials Pn(x ; q) ∈ T (T ={Pn(x ; q) ∈ Askey–Wilson polynomials: Al-Salam-Carlitz I, Discrete q-Hermite I, Little (Big) q-Laguerre, Little (Big) q-Jacobi, q-Hahn, Alternative q-Charlier) of any degree and for any order in terms of Pi(x ; q) themselves are proved. We will also provide two other interesting formulae to expand the coefficients of general-order q-difference derivatives Dqpf(x), and for the moments xℓDqpf(x), of an arbitrary function f(x) in terms of its original expansion coefficients. We used the underlying formulae to relate the coefficients of two different polynomial systems of basic hypergeometric orthogonal polynomials, belonging to the Askey–Wilson polynomials and Pn(x ; q) ∈ T. These formulae are useful in setting up the algebraic systems in the unknown coefficients, when applying the spectral methods for solving q-difference equations of any order.

In this paper we provide properties—which are, to the best of our knowledge, new—of the zeros of the polynomials belonging to the q -Askey scheme. These findings include Diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted. [ABSTRACT FROM AUTHOR]

Infinitely many Casoratian identities are derived for the Wilson and Askey–Wilson polynomials in parallel to the Wronskian identities for the Hermite, Laguerre and Jacobi polynomials, which were reported recently by the present authors. These identities form the basis of the equivalence between eigenstate adding and deleting Darboux transformations for solvable (discrete) quantum mechanical systems. Similar identities hold for various reduced form polynomials of the Wilson and Askey–Wilson polynomials, e.g. the continuous q -Jacobi, continuous (dual) ( q -)Hahn, Meixner–Pollaczek, Al-Salam–Chihara, continuous (big) q -Hermite, etc. [ABSTRACT FROM AUTHOR]

For a two-parameter family of lower triangular matrices with entries involving Jacobi polynomials an explicit inverse is given, with entries involving a sum of two Jacobi polynomials. The formula simplifies in the Gegenbauer case and then one choice of the parameter solves an open problem in a recent paper by Koelink, van Pruijssen & Román. The two-parameter family is closely related to two two-parameter groups of lower triangular matrices, of which we also give the explicit generators. Another family of pairs of mutually inverse lower triangular matrices with entries involving Jacobi polynomials, unrelated to the family just mentioned, was given by J. Koekoek & R. Koekoek (1999). We show that this last family is a limit case of a pair of connection relations between Askey–Wilson polynomials having one of their four parameters in common. [ABSTRACT FROM AUTHOR]

We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula [ABSTRACT FROM AUTHOR]

We give a general expansion formula of functions in the Askey–Wilson polynomials and using Askey–Wilson orthogonality we evaluate several integrals. Moreover we give a general expansion formula of functions in polynomials of Askey–Wilson type, which are not necessarily orthogonal. Limiting cases give expansions in little and big q -Jacobi type polynomials. We also give a new generating function for Askey–Wilson polynomials and a new evaluation for specialized Askey–Wilson polynomials. [ABSTRACT FROM AUTHOR]

New formulas for the $n^{\mathrm{th}}$ moment $\mu_n(a,b,c,d;q)$ of the Askey-Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured.

Abstract: New formulas for the nth moment of the Askey–Wilson polynomials are given. These are derived using analytic techniques, and by considering three combinatorial models for the moments: Motzkin paths, matchings, and staircase tableaux. A related positivity theorem is given and another one is conjectured. [Copyright &y& Elsevier]

The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a generalization of Askey–Wilson polynomials is introduced. These polynomials are obtained from the Askey–Wilson polynomials via the addition of two mass points to the weight function of them at the points ±1. Several properties of such new family are considered, in particular, the three-term recurrence relation and the representation as basic hypergeometric series. [Copyright &y& Elsevier]

Abstract: The linearization problem is the problem of finding the coefficients in the expansion of the product of two polynomial systems in terms of a third sequence of polynomials , The polynomials , and may belong to three different polynomial families. In the case , we get the (standard) linearization or Clebsch–Gordan-type problem. If , we are faced with the so-called connection problem. In this paper, we compute explicitly, in a more general setting and using an algorithmic approach, the connection and linearization coefficients of the Askey–Wilson orthogonal polynomial families. We find our results by an application of computer algebra. The major algorithmic tool for our development is a refined version of q-Petkovšekʼs algorithm published by Horn (2008), Horn et al. (2012) and implemented in Maple by Sprenger (2009), Sprenger and Koepf (2012) (in his package qFPS.mpl) which finds the q-hypergeometric term solutions of q-holonomic recurrence equations. The major ingredients which make this application non-trivial are [•] the use of appropriate operators and ; [•] the use of an appropriate basis for these operators; [•] and a suitable characterization of the classical orthogonal polynomials on a non-uniform lattice which was developed very recently (Foupouagnigni et al., 2011). Without this preprocessing the relevant recurrence equations are not accessible, and without the mentioned algorithm the solutions of these recurrence equations are out of reach. Furthermore, we present an algorithm to deduce the inversion coefficients for the basis in terms of the Askey–Wilson polynomials. Our results generalize and extend known results, and they can be used to deduce the connection and linearization coefficients for any family of classical orthogonal polynomial (including very classical orthogonal polynomials and classical orthogonal polynomials on non-uniform lattices), using the fact that from the Askey–Wilson polynomials, one can deduce, by specialization and/or by limiting processes, any other family of classical orthogonal polynomials. As illustration, we give explicitly the connection coefficients of the continuous q-Hahn, q-Racah and Wilson polynomials. [Copyright &y& Elsevier]

Previously, we proved a q-series expansion formula which allows us to recover many important classical results for q-series. Based on this formula, we derive a new q-formula in this paper, which clearly includes infinitely many q-identities. This new formula is used to give a new proof of the orthogonality relation for the Askey-Wilson polynomials. A curious q-transformation formula is proved, and many applications of this transformation to Hecke type series are given. Some Lambert series identities are also derived. [ABSTRACT FROM AUTHOR]

We explain how the moments of the (weight function of the) Askey-Wilson polynomials are related to the enumeration of the staircase tableaux introduced by the first and fourth authors. This gives us a direct combinatorial formula for these moments, which is related to, but more elegant than the formula given in their earlier paper. Then we use techniques developed by Ismail and the third author to give explicit formulae for these moments and for the enumeration of staircase tableaux. Finally we study the enumeration of staircase tableaux at various specializations of the parameterizations; for example, we obtain the Catalan numbers, Fibonacci numbers, Eulerian numbers, the number of permutations, and the number of matchings. [ABSTRACT FROM AUTHOR]

Abstract: We give a general method of characterizing symmetric orthogonal polynomials through a certain type of connection relations. This method is applied to Al-Salam–Chihara, Askey–Wilson, and Meixner–Pollaczek polynomials. This characterization technique unifies and extends some previous characterization results of Lasser and Obermaier and Ismail and Obermaier. Along the way we explicitly evaluate the connection coefficients in the expansion of in terms of , where is the Askey–Wilson operator and are general Askey–Wilson polynomials. As a limiting case we derive the corresponding connection coefficients in the expansion of in terms of , where is the Wilson operator and are general Wilson polynomials. Using the connection relation for Askey–Wilson polynomials, we obtain a characterization for the two-parameter symmetric Askey–Wilson polynomials. The connection relations between , and are also derived. [Copyright &y& Elsevier]